We study a particular \((1+2n)\)-body problem, conformed by a massive body and 2n equal small masses, since this problem is related with Maxwell’s ring solutions, we call planet to the massive body, and satellites to the other 2n masses. Our goal is to obtain doubly-symmetric orbits in this problem. By means of studying the reversing symmetries of the equations of motion, we reduce the set of possible initial conditions that leads to such orbits, and compute the 1-parameter families of time-reversible invariant tori. The initial conditions of the orbits were determined as solutions of a boundary value problem with one free parameter, in this way we find analytically and explicitly a new involution, until we know this is a new and innovative result. The numerical solutions of the boundary value problem were obtained using pseudo arclength continuation. For the numerical analysis we have used the value of \(3.5 \times 10 ^{-4}\) as mass ratio of some satellite and the planet, and it was done for \(n=2,3,4,5,6\). We show numerically that the succession of families that we have obtained approach the Maxwell solutions as n increases, and we establish a simple proof why this should happen in the configuration.

我们研究一个特定的\（（（1 + 2n）\）-物体问题，该问题由一个大质量物体和2 n个相等的小质量组成，因为该问题与麦克斯韦环解决方案有关，因此我们将行星称为大质量物体，将卫星称为其他2 n群众。我们的目标是在这个问题上获得双对称轨道。通过研究运动方程的反向对称性，我们减少了导致此类轨道的可能初始条件的集合，并计算了时间可逆不变花托的1参数族。轨道的初始条件被确定为具有一个自由参数的边值问题的解决方案，通过这种方式，我们可以分析并明确地发现一个新的对合，直到我们知道这是一个新的创新结果。使用伪弧长连续获得了边值问题的数值解。为了进行数值分析，我们使用\（3.5 \ times 10 ^ {-4} \）的值作为某些卫星与行星的质量比，并针对\（n = 2,3,4,5， 6 \）。我们用数字显示出，随着n的增加，获得的族系列接近Maxwell解，并且我们建立了一个简单的证明来证明为什么在配置中会发生这种情况。